Summary: PLASTIC TORSION AND RELATED PROBLEMS
FRANC¸OIS ALOUGES AND ANTONIO DESIMONE
Abstract. A simple algorithm to calculate the maximum torsional load for a
cylindrical shaft is presented. The algorithm is based on the notion of viscosity
solutions to the eikonal equation, and is not restricted to simply­connected
cross­sections. Applications to other, related problems, such as ferromagnetic
thin films, and elastic buckling of thin film blisters are also discussed.
1. Introduction
A classical problem in Plasticity (more specifically, in Limit Analysis) is that
of determining the maximum admissible torsional load for a cylindrical shaft. If
the yield criterion is von Mises', and if the cross section is simply connected, the
solution can be obtained through the ``sand heap'' analogy (see, e.g., [5]). When
the cross­section is not simply connected, the sand heap construction can still be
used, provided that constant heights are judiciously assigned on the contours of the
internal holes 1 .
In this note, we present a short algorithm to solve this problem, which is based
on the notion of viscosity solution to the eikonal equation. The key observation
is that the viscosity solution is the maximal pointwise Lipschitz solution to the
eikonal equation, i.e., the tallest admissible sand heap.
We also discuss the application of the algorithm to a few more physical systems,